A thermodynamically consistent phase field approach for fracture including surface stresses is presented. The surface stresses which are distributed inside a finite width layer at the crack surface are introduced as a result of employing some geometrical nonlinearities, i.e., by defining energy terms per unit volume at the current time and evaluating gradient of the order parameter in the current configuration. A double-well barrier term is included in the structure of the Helmholtz free energy which allows one to use the energy terms per unit volume at the current time when introducing the surface stresses in the current approach. Thus, the surface stresses are introduced in a similar way to the interfacial stresses in phase transformations. The differences in the modeling of crack growth with considering the surface stresses and without it are discussed. It is shown how the surface stresses affect the stress fields and consequently the crack nucleation and propagation. The finite element method is utilized to solve the coupled equations of mechanics and crack phase field. It is emphasized that the surface stresses affect the driving force for both the crack nucleation and propagation by disturbing the momentum balance. Thus, a different external loading is required in the presence of the surface stresses.

中文翻译：

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具有双阱势的裂纹扩展相场模型，包括表面效应

提出了一种热力学一致的相场方法，用于断裂，包括表面应力。由于采用了一些几何非线性，即通过在当前时间定义每单位体积的能量项并在当前配置中评估阶数参数的梯度，引入了在裂纹表面的有限宽度层内部分布的表面应力。 。亥姆霍兹自由能的结构中包含一个双阱势垒项，当在当前方法中引入表面应力时，该势垒项允许当前使用每单位体积的能量项。因此，以与相变中的界面应力相似的方式引入表面应力。讨论了在考虑表面应力和不考虑表面应力的情况下裂纹扩展建模的差异。结果表明，表面应力如何影响应力场，进而影响裂纹的形核和扩展。利用有限元方法求解力学和裂纹相场的耦合方程。要强调的是，表面应力通过扰动动量平衡来影响裂纹成核和扩展的驱动力。因此，在存在表面应力的情况下需要不同的外部载荷。要强调的是，表面应力通过扰动动量平衡来影响裂纹成核和扩展的驱动力。因此，在存在表面应力的情况下需要不同的外部载荷。要强调的是，表面应力会通过扰动动量平衡来影响裂纹成核和扩展的驱动力。因此，在存在表面应力的情况下需要不同的外部载荷。